After weeks marked by technological breakdowns, regulatory issues and general questions over its viability, bitcoin is in the midst of the worst crisis since it was proposed in a white paper in 2008.

The turmoil is a watershed. The way bitcoin and its ecosystem react to the problems could determine whether the whole experiment goes the way of Dutch tulips in the 1600s or becomes a historic technological breakthrough like email.

To recap: Bitcoin is a virtual or “crypto” currency. It isn’t backed by governments or central banks and it exists only online. In fact, it is so “crypto” that nobody knows who developed it.

There is no mint either. Users “mine” bitcoins by using computer algorithms to solve complex mathematical problems. They trade the coins among themselves or spend them on things like tickets to Sacramento Kings basketball games, or at online retailer Overstock.com.

Over the past three years, bitcoin has emerged from geekland into the mainstream. Its price skyrocketed from a few cents in 2011 to a high of more than $1,100 in December 2013. More and more merchants started accepting it. And its most ardent friends and foes even predicted that it might one day replace the dollar and other traditional currencies.

Bitcoin’s strength is predicated on three supposed qualities: It is anonymous, or at least pseudonymous (transactions are recorded but the identity of the parties is encrypted); it is difficult to hack; and it cuts out financial middlemen like banks.

Ajay Banga, chief executive of Mastercard Inc., spoke for many skeptics last week when he said in an interview: “The world is not short of currencies, so what is this currency solving for?”

That is the crucial question. In order to thrive, bitcoin has to be more useful than current payment systems.

Two possible applications could embed virtual currencies into the financial infrastructure in a way that is complementary to existing forms of payments.

The first is as conduits for small international transactions like remittances from foreign workers. Bitcoin could reduce both the cost and the time required for such payments.

For Antonis Polemitis, this means that consumers should rejoice, and money transmission firms like Western Union Co. and MoneyGram International Inc. should worry. “Bitcoin is like email and the other stuff is like the post office,” says Mr. Polemitis, who heads Ledra Capital LLC, a family office looking to invest in the bitcoin arena.

Supporters like Mr. Malka view bitcoin as more than just a currency used to store value. For them, bitcoin is an open financial platform that could house scores of types of data in a secure and universal ledger. From payments for road tolls to proof of ownership for cars and houses, bitcoin could be an independent, secure and reliable host of financial and personal information.

“It’s not a technology experiment, it is an experimental society,” Mr. Malka says.

It sounds far-fetched. Even bitcoin evangelists concede that this could only happen if three conditions materialize.

First, the current infrastructure—largely anonymous, anchored by unregulated overseas exchanges and vulnerable to manipulation by criminals—must be overhauled through the creation of U.S. exchanges overseen by financial watchdogs. Second, institutional money like pension funds will have to invest in bitcoin to curb its wild price volatility. And, third, banks will have to view bitcoin as legitimate and enable customers to exchange it for dollars and cents.

None of those is a given, and none of those would make bitcoin a replacement for the dollar. But behind all the hype and fear, there is a real possibility that bitcoin will play more than a bit part in the financial industry of the future.